An integrated model for stock replenishment and shipment scheduling under common carrier dispatch costs

Transportation Research Part E 46 (2010) 844–854

Contents lists available at ScienceDirect

Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

An integrated model for stock replenishment and shipment schedulingunder common carrier dispatch costs q

Fatih Mutlu a, Sıla Çetinkaya b,*

a Department of Mechanical and Industrial Engineering, Qatar University, Doha 2713, Qatarb Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843-3131, United States

a r t i c l e i n f o

Article history:Received 22 October 2009Received in revised form 27 January 2010Accepted 16 March 2010

Keywords:Shipment consolidationVendor-managed inventoryCommon carrier

1366-5545/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.tre.2010.05.001

q This research was supported in part by NSF Gra* Corresponding author.

E-mail addresses: [email protected] (F. Mutl

a b s t r a c t

We examine a joint inventory replenishment and shipment scheduling problem that arisesin the context of a vendor-managed inventory (VMI) arrangement. Since a temporal ship-ment consolidation policy is being implemented, the inventory requirements at the vendorare affected by the timing and quantity of shipment release. The vendor’s problem is todetermine an integrated policy for inventory replenishment and shipment release and toset its parameters. We develop analytical models for computing such integrated policieswhere it is economical to use common carriage for outbound transportation. We proposealgorithmic approaches to set the optimal policy parameter values.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction and related literature

We revisit a joint inventory replenishment and shipment scheduling problem originally introduced in Çetinkaya and Lee(2000). The problem arises in the context of a vendor-managed inventory (VMI) arrangement (Çetinkaya and Lee, 2000;Dong et al., 2007; Yao and Dresner, 2008) where the vendor has the authority to control the supply of an item for a groupof retailers located in close proximity to each other, i.e., in a given market area. The retailers do not carry inventory for theitem except for display models (such as personal computers, household appliances, photocopy machines, or bulky furniture).Each customer demand is transmitted to the vendor as a retailer order via the use of electronic data interchange (EDI) tech-nology. The time between each customer order at each retailer is stochastic. Thus, the timing of retailer orders at the supplieris also stochastic. Utilizing the information shared via EDI, the vendor decides on the size and timing of resupply, i.e., out-bound dispatch, to the retailers. Hence, the vendor may consolidate orders from the retailers in the market area into a largershipment and benefit from the scale economies associated with its outbound transportation costs. This practice is known astemporal shipment consolidation in the context of outbound dispatch under VMI.

In this setting, the inventory requirements at the vendor are affected by the timing, as well as the quantity, of shipmentrelease, i.e., retailer resupply, decisions. Then, the vendor’s problem in Çetinkaya and Lee (2000) is to determine an integratedpolicy for inventory replenishment and shipment release and to set its parameters. Hence, the corresponding model is calledan integrated model.

Two types of temporal shipment consolidation routines are popular and have been studied in the context of VMI appli-cations: (i) time-based and (ii) quantity-based shipment consolidation policies. A time-based policy ships accumulated loads(clears all outstanding orders) every T periods whereas a quantity-based policy ships an accumulated load when an econom-ical dispatch quantity, say q, is available. A third class of policies is known as hybrid (or time-and-quantity) policies under

. All rights reserved.

nt CAREER/DMI-0093654.

u), [email protected] (S. Çetinkaya).

http://dx.doi.org/10.1016/j.tre.2010.05.001

mailto:[email protected]

mailto:[email protected]

http://www.sciencedirect.com/science/journal/13665545

http://www.elsevier.com/locate/tre

F. Mutlu, S. Çetinkaya / Transportation Research Part E 46 (2010) 844–854 845

which a consolidated shipment is released either when q is accumulated or when the waiting time of an order exceeds Tbefore q is consolidated.

The integrated model in Çetinkaya and Lee (2000) examines the case where the vendor implements a time-based policy.Specifically, the vendor faces a Poisson order process and implements a time-based consolidation policy, with a shipmentrelease (outbound dispatch) interval of T, for dispatching consolidated retailer orders, where T is a decision variable alongwith the vendors order-up-to-level, denoted by Q. The problem is then the simultaneous computation of the optimal Tand Q values so as to minimize the expected long-run average cost. For this problem, an approximate solution approachis developed in Çetinkaya and Lee (2000) and an exact solution procedure is provided later in Axsäter (2001). Recently,the results in Çetinkaya and Lee (2000) are extended in Çetinkaya et al. (2006) to study the cases where the vendor imple-ments quantity-based and hybrid policies, and, also, in Çetinkaya et al. (2008) to consider general bulk demand processesunder quantity-based policies.

All of these existing integrated models assume private carriage associated with outbound dispatches. This assumption, inturn, implies that the transportation cost to the market area consists of a fixed cost per dispatch and a unit cost associated witheach item in the load. Such a cost structure may also be appropriate for modeling contract carriage. Private and contract carriageare often used when shipment size is large enough to justify a full-truck-load (FTL) rate. However, as noted in Chan et al. (2002),‘‘in the retail, grocery, and electronic industries, the most frequently used mode of transportation is the less-than-truckload(LTL) mode, which is attractive when shipment sizes are less than the truck capacity.” Our focus in this paper pertains to suchsituations where it is economical to use common carriage, rather than a private fleet of vehicles, for outbound transportation.

Specifically, revisiting the original problem setting in Çetinkaya and Lee (2000), we examine the case where the vendoruses common carriage for outbound dispatches in which case the corresponding transportation cost assumes a significantlymore complex form. Hence, the results in this paper generalize the existing literature to incorporate a practical, general, andcomplicated dispatch cost structure. Our goal is to apply cost-based optimization for the purpose of computing the param-eters of integrated policies for the vendor’s problem at hand considering both time-based and a quantity-based policies forshipment consolidation. It is worthwhile to note that, in the context of the VMI arrangement of interest in this paper, ship-ment consolidation is still a viable option under common carriage because the corresponding transportation costs also ex-hibit scale economies, i.e., larger volumes are shipped at discounted rates (Higginson, 1993; Ballou, 2003).

The shipment consolidation literature until 2000 focuses exclusively on pure consolidation practices, i.e., shipment consol-idation only in isolation from other logistical decisions. The literature on pure consolidation (i) compares the alternative pol-icies mentioned above using simulation (Closs and Cook, 1987; Cooper, 1984; Higginson and Bookbinder, 1994; Jackson,1981; Masters, 1980), (ii) investigates the justification and evaluation of such practices (Abdelwahab and Sargious, 1990;Burns et al., 1985; Campbell, 1990; Daganzo, 1988; Hall, 1987; Pollock, 1978), and (iii) proposes optimization approaches(Çetinkaya and Bookbinder, 2003; Gupta and Bagchi, 1987; Higginson and Bookbinder, 1995; Mutlu et al., 2010) for settingthe optimal consolidation policy parameters in isolation of their effect on inventory policy parameters.

Our focus in this paper lies within the more recent line of research which extends the literature on pure consolidation byanalyzing the impact of alternative consolidation policies in the context of joint inventory and transportation decisions. Thisline of work focuses on integrated models that rely on cost-based optimization for the purpose of computing the parametersof inventory and transportation policies, simultaneously (Çetinkaya and Lee, 2000; Çetinkaya et al., 2006; Çetinkaya et al.,2008). As we noted earlier, the main argument motivating these integrated models is that the timing and quantity of resupplyto the retailers as implied by the shipment release practice in place impose a dependency between inventory and transpor-tation decisions which, in turn, should be optimized simultaneously. Clearly, this argument is consistent with the idea of inte-grating various logistical decisions and entities in the supply chain which has been setting the major trends in recent practicesto include not only VMI, but, also, third-party logistics (3PL), channel coordination, and contractual procurement applications.

We proceed with detailed descriptions of the problem setting and common carrier dispatch costs of interest in Section 2.Next, in Section 3, we study the case where a time-based policy is in place for shipment consolidation. This is followed by, inSection 4, an analysis of the case where a quantity-based policy is used. Our concluding comments are provided in Section 5.

2. Modeling assumptions and the dispatch cost structure

Each retailer, indicated by i, observes a stochastic stream of customer demand, characterized by a Poisson process withparameter ki, for a certain type of product. That is, the time between the demands for customer i is distributed according toan exponential distribution with parameter ki and each customer requests exactly 1 unit. Under the assumptions of the VMIarrangement of interest, the retailers do not carry inventory and every time a customer arrives, the demand information istransmitted to the vendor. Then, the vendor observes an order process, which is a superposition of the retailers’ demand.That is, the order process at the vendor is also Poisson with parameter k ¼

Pki. As we noted in the previous section, the

vendor has the flexibility of managing the timing and quantity of resupply in response to these orders, and a common carrieris used for order shipments. In order to benefit the scale economies inherent in this context, the vendor wishes to consolidateorders to bundle them into a larger load. As part of the VMI agreement under consideration, the vendor compensates theretailers for order delivery delays due to consolidation. This compensation is represented by a order waiting cost term,denoted by w, which is incurred on a per unit per time basis. The vendor’s challenge is to balance the trade-off betweenthe transportation scale economies and penalty of delaying the order shipments, both due to shipment consolidation.

846 F. Mutlu, S. Çetinkaya / Transportation Research Part E 46 (2010) 844–854

The vendor operates a warehouse for carrying inventory, eventually, to be shipped to the market area. A per unit holdingcost of h is incurred for each item in stock for a unit time at the warehouse. The warehouse inventory is by supplied an exter-nal source, say, a distributor, with ample supply and negligible lead time, at a fixed cost of AR and a unit cost of cR. Hence,under the cost and demand assumptions of the setting of interest, it suffices to consider the policy given by (s = �1,S = Q) forinventory replenishment decisions at the warehouse so that there is no risk of supply shortage at the warehouse. Under thispolicy whenever the inventory level hits s = �1 or below, an replenishment order is placed by the vendor to the distributor,immediately raising the warehouse inventory up to S = Q. Since we are interested in the corresponding integrated modelsunder both time-based and quantity-based shipment consolidation, we deal with two problems. The first problem pertainsto the simultaneous computation of the parameter values of the integrated policy characterized by the inventory replenish-ment policy with parameters (s = �1,S = Q) and time-based consolidation policy with parameter T. The second problem issimilar, with the exception that we are interested in computing the quantity-based consolidation policy parameter q. Onecan easily observe that two main flows are associated with our model. One of them is from vendor to the market area,and the other is from distributor to the vendor. The former represents the outbound flow whereas the latter representsthe inbound flow.

Under the time-based policy we consider here, the main consideration is the time between two consecutive dispatches.The vendor consolidates the retailer orders and then makes a dispatch decision after T units of time have elapsed since thelast dispatch. In other words, vendor dispatches to the retailer every T time units. Under the quantity-based policy we exam-ine, the vendor makes a dispatch decision when the consolidated orders reach up to certain quantity, q.

In our model, the vendor uses a common carrier to dispatch the items to the retailer. The common carrier offers an allunits discount schedule based the quantity shipped. This dispatch cost structure is the main difference of our model whencompared to the integrated models in Çetinkaya and Lee (2000) and Çetinkaya et al. (2006). We explain the cost structure indetail later in this section.

Our goal is to find the optimal inventory replenishment and consolidation policy parameters that minimize the expectedlong-run average cost of the vendor. This cost includes the replenishment cost, inventory holding cost, dispatch cost and cus-tomer waiting cost. We note that a summary of the notation commonly used throughout the paper is presented in Table 1,and we proceed with a discussion of the dispatch cost structure of interest.

As mentioned above, vendor uses a common carrier to dispatch the items. The common carrier rates as a function of theshipment quantity, q, is given as

Table 1Basic n

Sym

AR

AD

cR

hwkTneQQ

Tt

CðeQKqkC(k,qC*

E[HCE[WCE[RCE[DCF(�)p(�)

cD ¼

c0 if 1 6 q < q2;

c1 if q2 6 q < q4;

..

.

ci if q2i 6 q < q2iþ2;

..

.

cI if q2I 6 q:

8>>>>>>>>>><>>>>>>>>>>:
otation.
bol Definition

Fixed cost of replenishing vendor’s inventoryFixed cost of an outbound dispatchUnit procurement costVendor’s inventory carrying cost per unit timeCustomer waiting penalty per unit timeRate of Poisson arrival of ordersTime between the nth and (n � 1)th orders following an exponential distribution with mean 1/kOrder-up-to-level at the vendor

By definition, Q ¼ eQ þ 1Dispatch (consolidation) cycle length under time-based policyBy definition, t = kT

; tÞ Long-run average of expected total cost under time-based policy

Number of dispatch cycles within a replenishment cycle under time-based policyDispatch quantity in quantity-based policyNumber of dispatch cycles within a replenishment cycle under quantity-based policy. By definition, Q = (k � 1)q

) Long-run average of expected total cost under quantity-based policyOptimal cost

] Expected holding cost in a replenishment cycle] Expected customer waiting cost in a replenishment cycle

] Expected replenishment cost in a replenishment cycle] Expected dispatch cost in a replenishment cycle

Poisson distribution functionPoisson density function


Under this freight rate schedule, the dispatch cost, DC(q), for the vendor is

DCðqÞ ¼

c0q if 1 6 q < q2;

c1q if q2 6 q < q4;

..

.

ciq if q2i 6 q < q2iþ2;

..

.

cIq if q2I 6 q:

8>>>>>>>>>><>>>>>>>>>>:
Fig. 1 displays an example for DC(q) for I = 3. Even though the listed price suggests the above dispatch cost for the vendor, theactual cost will be different because of overdeclaration, which is a logistics practice that lets the shipper to declare a largerload than actual to get the lower rates. We explain why overdeclaration happens with an example. Assume that the unitfreight rate is $10 for 1 6 q < 20; and 8 for 20 6 q < 40. With this schedule, the cost of shipping 19 units is $190. Howeverinstead of declaring 19 units, the shipper may declare the load to be 20 units and get the lower rate so that the total costbecomes $160. In fact, overdeclaration leads to lower costs when the shipment load is between 16 and 20. Hence, the totaldispatch cost for quantities between 16 and 20 becomes a constant amount. As a consequence of overdecleration, the real-ized dispatch cost function is
DCðqÞ ¼

c0q if 1 6 q < q1;

c0q1 if q1 6 q < q2;

..

.


ciq2iþ1 if q2iþ1 6 q < q2iþ2;

..

.

cIq if q2I 6 q:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:
Fig. 2 shows how DC(q) changes after overdeclaration. At this point, we note that q2i+1’s are not necessarily integer.
3. Integrated policy under time-Based shipment consolidation

The model we examine in this section is an extension of the one in Çetinkaya and Lee (2000) to consider common carrierdispatch costs. As in Çetinkaya and Lee (2000), we also use the renewal reward theorem to compute the long run average ofthe expected total cost. The decision variables of the model under the time-based policy are

Fig. 1. DC(q) before overdeclaration.

Fig. 2. DC(q) after overdeclaration.


eQ : the order-up-to-level for replenishing the inventory, andT: the dispatch (consolidation) cycle length, i.e., the time between two dispatches.

For notational convenience, we let t = kT, and we use t as an indicator for the dispatch cycle length in the rest of the paper.In this setting, we note that each inventory replenishment epoch at the vendor is a regeneration epoch and we define a

replenishment cycle as the time between two consecutive replenishment epochs. Using the renewal reward theorem, theaverage cost is the ratio of the expected replenishment cycle cost to the expected replenishment cycle length. Each replen-ishment cycle incurs several cost components: expected replenishment cost denoted by E[RC], expected inventory carryingcost denoted by E[HC], expected dispatch cost denoted by E[DC], and expected customer waiting cost denoted by E[WC].Then, the expected replenishment cycle cost is expressed as E[RC] + E[HC] + E[DC] + E[WC], whereas the cost function ofinterest is given by

CðeQ ; tÞ ¼ E½RC� þ E½HC� þ E½DC� þ E½WC�E½Replenishment Cycle Length� : ð1Þ

For our model, all of the components in Expression (1) except for E[DC] are the same as in Çetinkaya and Lee (2000). Due tothe difference in the common carrier costs in outbound dispatch, we provide a derivation of E[DC] in the following section,and refer to Çetinkaya and Lee (2000) for the other components of Expression (1).

3.1. Derivation of E[DC] under time-based shipment consolidation

Before going into the details of the calculation, we note that we treat the dispatch quantity, q, as a continuous variable foranalytical tractability so that the following property holds:

ciq2iþ1 ¼ ciþ1q2ðiþ1Þ:

We define the time between two dispatches as a dispatch cycle. For the time-based policy, the number of dispatch cycleswithin a replenishment cycle is a random variable denoted by K. In order to compute E[DC], we first compute the expecteddispatch cost for one dispatch cycle and then multiply it by E[K]. (Derivation of E[K] is given in Çetinkaya and Lee (2000).) Theexpected dispatch cost for one dispatch cycle, E[DCC], is given by

E½DCC� ¼ c0

Xq1

x¼1

xpðxÞ þ c0q1

Xq2

x¼q1þ1

pðxÞ þ . . .þ ci

Xq2iþ1

x¼q2iþ1

xpðxÞ

þ ciq2iþ1

Xq2iþ2

q2iþ1þ1

pðxÞ þ . . .þ cI

X1x¼q2Iþ1

xpðxÞ;


where p(x) = (t)xe�t/x!.

E½DCC� ¼XI�1

i¼0

ci

Xq2iþ1

x¼q2iþ1

xtxe�t

x!þ ciq2iþ1ðFðq2iþ2Þ � Fðq2iþ1ÞÞ

!þ cI

X1x¼2Iþ1

xtxe�t

x!

¼XI�1

i¼0

citXq2iþ1�1

x¼q2i�1

txe�t

x!þ ciq2iþ1ðFðq2iþ2Þ � Fðq2iþ1ÞÞ

!þ cIt

X1x¼2I

txe�t

x!

¼XI�1

i¼0

ðcitðFðq2iþ1 � 1Þ � Fðq2i � 1ÞÞ þ ciq2iþ1ðFðq2iþ2Þ � Fðq2iþ1ÞÞÞ þ cItð1� Fðq2I � 1ÞÞ

¼ tXI�1

i¼0

ciðFðq2iþ1 � 1Þ � Fðq2i � 1ÞÞ þXI

i¼1

ciq2iðFðq2iÞ � Fðq2i�1ÞÞ þ cItð1� Fðq2I � 1ÞÞ:

Then, E[DC] is simply obtained by multiplying the above with E[K].

3.2. Analysis of the cost function C(Q, t) and the solution

We revisit the following results of Çetinkaya and Lee (2000) for finalizing the derivation of the cost function:

E½RC� ¼ AR þ cRE½K�t;

E½HC� ¼ heQ tkþ heQ ðeQ þ 1Þ

2k;

E½WC� ¼ wt2

2kE½K�; and

E½Replenishment Cycle Length� ¼ E½K�tk

:

We note that an exact expression of E[K] is also provided in Çetinkaya and Lee (2000) but it does not yield a closed-form. Foranalytical tractability, the following closed-form approximation is provided in Çetinkaya et al. (2006):

E½K� �1 if eQ þ 1 6 t;eQþ1

t if Q þ 1 > t;

8<: :

It is also shown in Çetinkaya et al. (2006) that this approximation performs very well for all practical purposes, i.e., it leads toresults very similar to the exact expression of E[K]. Hence, we also use this approximation. Combining our derivation of E[DC]with the above results in Expression (1), we then have

CðeQ ; tÞ ¼

ARkt þ cRkþ heQ þ heQ ðeQþ1Þ

2t

þkPI�1

i¼0ciðFðq2iþ1 � 1Þ � Fðq2i � 1ÞÞ

þkPI

i¼1

ciq2it ðFðq2iÞ � Fðq2i�1ÞÞ

þkcIð1� Fðq2I � 1ÞÞ þ wt2 ; if eQ þ 1 6 t;

ARkeQþ1þ cRkþ hteQeQþ1

þ heQ2

þkPI�1


þkPI

i¼1


þkcIð1� Fðq2I � 1ÞÞ þ wt2 ; if eQ þ 1 > t:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2Þ

Clearly, if eQ þ 1 6 t then there is only one dispatch cycle within a given replenishment cycle. In this case, it is optimal tosynchronize the replenishment of inventory and the outbound dispatch. That is, since the inventory replenishment lead timeis 0, it is then optimal for the vendor to delay the inventory replenishment until the time of the dispatch. Thus, no inventoryis carried at the vendor, i.e., eQ ¼ 0. Using this observation, Expression (2) can be simplified further. We make a change ofnotation by defining Q ¼ eQ þ 1 P 1 and express the modified cost function as


CðQ ; tÞ ¼

ARkt þ cRk

þkPI�1


þkPI

i¼1


þkcIð1� Fðq2I � 1ÞÞ þ wt2 ; if t P 1;

ARkQ þ cRkþ ht � ht

Q þhðQ�1Þ

2

þkPI�1


þkPI

i¼1


þkcIð1� Fðq2I � 1ÞÞ þ wt2 ; otherwise:

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

ð3Þ

Then, our problem is

min CðQ ; tÞs:t: Q 2 Nþ; t P 0:

Since we cannot know the value of Q beforehand, we have to solve the two cases separately: Single Dispatch Case (Q = 1) andMulti Dispatch Case (Q > 1).

Since cRk is constant, we do not consider this cost component in the rest of the paper.

3.2.1. Single dispatch caseThe case of single dispatch (within a replenishment cycle) implies that no actual inventory is carried at the vendor. The

vendor waits for the retailers’ orders and when a sufficient amount of time has passed since the last dispatch, the vendorcreates a replenishment order whose amount is equal to the number of waiting retailer orders. The replenishment ordersarrive in zero lead time and then are directly dispatched to the retailers. This is called a pure consolidation setting in theliterature. In the pure shipment consolidation setting, the objective function depends on only one decision variable, t. Hence,we use C(t) instead of C(Q, t). Recalling from Expression (3), we have

CðtÞ ¼ ARktþ k

XI�1

i¼0

ciðFðq2iþ1 � 1Þ � Fðq2i � 1ÞÞ þ kXI

i¼1

ciq2i

tðFðq2iÞ � Fðq2i�1ÞÞ þ kcIð1� Fðq2I � 1ÞÞ þwt

2:

If the function is convex in the region where t P 1, then it suffices to find the t value that solves for the first derivative. Takingthe derivative of C(t), we have

dCðtÞdt¼ w

2� ARk

t2 � kXI�1

i¼0

ciðpðq2i � 1Þ � pðq2iþ1 � 1ÞÞ þ kt

XI

i¼1

ciq2iðpðq2i�1Þ � pðq2iÞÞ �k

t2

XI

i¼1

ciq2iðFðq2iÞ � Fðq2i�1ÞÞ

þ kcIpðq2I � 1Þ:

Then, using ciq2i+1 = ci+1q2(i+1) in the above expression, it is easy to show that

dCðtÞdt¼ w

2� ARk

t2 � kXI�1

i¼0

ciðpðq2i � 1Þ � pðq2iþ1 � 1ÞÞ þ kt

XI

i¼1

ci�1q2i�1pðq2i�1Þ �kt

XI

i¼1

ciq2ipðq2iÞ �k

t2

XI

i¼1

ciq2iðFðq2iÞ

� Fðq2i�1ÞÞ þ kcIpðq2I � 1Þ

¼ w2� ARk

t2 � kXI�1

i¼0

ciðpðq2i � 1Þ � pðq2iþ1 � 1ÞÞ þ kXI

i¼1

ci�1pðq2i�1 � 1Þ � kXI

i¼1

cipðq2i � 1Þ � k

t2

XI

i¼1

ciq2iðFðq2iÞ


¼ w2� ARk

t2 � kXI�1

i¼0

ciðpðq2i � 1Þ � pðq2iþ1 � 1ÞÞ þ kXI�1

i¼0

cipðq2iþ1 � 1Þ � kXI

i¼1

cipðq2i � 1Þ � k

t2

XI

i¼1

ciq2iðFðq2iÞ


¼ w2� ARk

t2 �k

t2

XI

i¼1

ciq2iðFðq2iÞ � Fðq2i�1ÞÞ:

To check convexity, we examine the second derivative given by


d2CðtÞdt2 ¼ 2ARk

t3 þ kXI

i¼1

2ciq2i

t3 ðFðq2iÞ � Fðq2i�1ÞÞ þciq2i

t2 ðpðq2iÞ � pðq2i�1ÞÞ� �

;

and observe that it is always positive when p(q2i) P p(q2i�1) for all i. In the case where p(q2i) > p(q2i�1) for some i, the suf-ficient condition for convexity is given by 2(F(q2i) � F(q2i+1))/p(q2i) � p(q2i�1) P t.

So far, we have given the sufficient conditions for convexity. Hence, the next step is to compute the value of T such thefirst derivative is equal to 0 and check to see whether the underlying function is convex at that point.

3.2.2. Multi dispatch caseIn this case, the cost function is given by

CðQ ; tÞ ¼ ARkQþ ht � ht

Qþ hðQ � 1Þ

2þ k

XI�1

i¼0

ciðFðq2iþ1 � 1Þ � Fðq2i � 1ÞÞ þ kXI

i¼1

ciq2i

tðFðq2iÞ � Fðq2i�1ÞÞ

þ kcIð1� Fðq2I � 1ÞÞ þwt2:

The cost function has two decision variables, Q and t. One way to minimize C(Q, t) is to look at the stationary points whereboth @C(Q, t)/@Q and @C(Q, t)/@t are 0. However, for such a point, if it exists, to be the minimizer, the function must be jointlyconvex in Q and t. This requires a Hessian check. Below we present the first and second order partial derivatives:

@CðQ ; tÞ@t

¼ h� hQþw

2� k

t2

XI

i¼1

ciq2iðFðq2iÞ � Fðq2i�1ÞÞ;

@CðQ ; tÞ@Q

¼ h2þ ht � ARk

Q 2 ;

@2CðQ ; tÞ@t2 ¼ k

XI

i¼1

2ciq2i

t3 ðFðq2iÞ � Fðq2i�1ÞÞ þciq2i

t2 ðpðq2iÞ � pðq2i�1ÞÞ� �

;

@2CðQ ; tÞ@Q 2 ¼ 2ðARk� htÞ

Q3 ; and

@2CðQ ; tÞ@Q@t

¼ h

Q 2 :

We compute the determinant of the Hessian matrix as

jHj ¼ kXI

i¼1

4ciq2iðARk� htÞðFðq2iÞ � Fðq2iþ1ÞÞQ 3t3

�þ ciq2i2ðARk� htÞðpðq2iþ2Þ � pðq2iþ1ÞÞ

Q 3t2

�� h2

Q4 : ð4Þ

From Expression (4), we conclude that C(Q, t) is not necessarily a jointly convex function. Note that C(Q, t) is not necessarilyconvex in t either. For this reason, we propose the following finite search algorithm over possible values of Q and t.

Algorithm 1

Step 0: Set Q = 0, C* = M, where M is a very large number.Step 1: Q = Q + 1 and tm

Q ¼ ðQ þ 1Þ.Step 2: If Q P Qs then STOP, else go to Step 3.Step 3: grid ¼ minf0:01; tm

Q =ð50kÞg.Step 4: for t = 0.01 to t 6 tm

Q , increase t by grid; evaluate C(Q, t).Step 5: If C(Q, t) < C*, set t* = t and Q* = Q. Go to Step 1.

The Qs value in this algorithm is the stopping Q value for the search. The following lemma helps determine Qs.

Lemma 1. A stopping value of Q, namely Qs, is

Q s ¼ffiffiffiffiffiffiffiffiffiffiffi2ARk

h

r:

Proof. Let Qs be a stopping value. Then for any Q = Qs + m, m P 0, C(Q,T) must be greater than C(Qs, t) for all t; in otherwords, C(Q, t) � C(Qs,t) P 0 where

CðQ ; tÞ � CðQs; tÞ ¼ hm2� ARk

mðQ s þmÞðQ sÞ

þ htm

ðQs þmÞðQ sÞ:

We guarantee that the above expression is nonnegative when h/2 P ARk/(Qs)2. This completes the proof. h


3.2.3. SolutionAfter obtaining the solutions of the Single Dispatch and Multi Dispatch cases, we compare both solutions and pick the one

with the minimum cost.

4. Integrated policy under quantity-based shipment consolidation

Similar to the approach we followed for the time-based policy, we first express the cost function in terms of the decisionvariables. The decision variables are Q, the inventory order-up-to-level, and q, the target consolidation quantity. Followingthe results of Çetinkaya et al. (2006), in the optimal solution, Q will be an integer multiple, say k, of q. Here, k denotes thenumber of dispatch cycles in a replenishment cycle and Q = (k � 1)q. By making a change of variables, we can express thecorresponding expected long-run average cost, denoted by C(k,q), as a function of k and q.

In order to derive the cost function, we mainly refer to Çetinkaya et al. (2006) except for the expected dispatch cost. Theapproach is the same as the previous case: computing the expected replenishment cycle cost and dividing it by E[Replenish-ment Cycle Length].

Next, we derive the expression for the expected dispatch cost under common carrier charges and then combine this resultwith those of Çetinkaya et al. (2006) to obtain the expression of the long-run average of the expected total cost.

4.1. Derivation of the cost function under quantity-based shipment consolidation

For the quantity-based policy, the dispatch quantity is no longer a random variable. Hence, we can express the expectedcost of a single dispatch as follows:

E½DCC� ¼

c0q if 1 6 q < q1;

c0q1 if q1 6 q < q2;

..

.


ciq2iþ1 if q2iþ1 6 q < q2iþ2;

..

.

cIq if q2I 6 q:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:
Then the expected dispatch cost for one replenishment cycle, E[DC], is simply given by multiplying the above expression bykE[Dispatch Cycle Length]. Note that, since E[DC] is a piecewise function, C(k,q), is also a piecewise function. Referring toÇetinkaya et al. (2006) for expressions of E[RC], E[HC], E[WC], and E[Replenishment Cycle Length], and combining theseexpressions with E[DC] that we computed above, we obtain the cost function as
Cðk; qÞ ¼

½hðk�1Þþw�q2 þ ARk

kq � w2 þ c0k if 1 6 q < q1;


kq � w2 þ

A0Dkq if q1 6 q < q2;

..

.


kq � w2 þ cik if q2i 6 q < q2iþ1;


kq � w2 þ

AiDkq if q2iþ1 6 q < q2iþ2;

..

.


kq � w2 þ cIk if q2I 6 q:

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:
where Ai
D ¼ ciq2iþ1.

4.2. Analysis of the cost function C(k,q) and the solution

We state our problem as:

min Cðk; qÞs:t: k; q 2 Nþ:

Although the cost function has multiple pieces, we can see that there are two types of functions, one with a fixed dispatchcost (having Ai

Dk=q term) and the other is without a fixed dispatch cost but with a per unit dispatch cost (having cik term). Wename these functions FCi(k,q) and LCi(k,q), respectively:


FCiðk; qÞ ¼ðhðk� 1Þ þwÞq

2þ ARk

kq�w

2þ Ai

Dkq;

LCiðk; qÞ ¼ðhðk� 1Þ þwÞq

2þ ARk

kq�w

2þ cik:

Hence, the resulting subproblems are:

min FCiðk; qÞ

s:t: q2iþ1 6 q < q2iþ2

k; q 2 Nþ;

min LCiðk; qÞ

s:t: q2i 6 q < q2iþ1

k; q 2 Nþ:

Observe that both FCi(k,q) and LCi(k,q) have a structure similar to the cost function studied in Çetinkaya et al. (2006). How-ever, we cannot simply follow the solution technique that was developed in Çetinkaya et al. (2006), because the our subprob-lems have finite upper and lower bounds on q. Next, we present some properties of FCi(k,q) and LCi(k,q).

4.2.1. Analysis of FC(k,q)Observe that, FCi(k,q) is convex in k. Momentarily assuming k as a continuous variable, we can easily show that for a given

q, the optimal k, namely k*(q) is

k�ðqÞ ¼ 1q

ffiffiffiffiffiffiffiffiffiffiffi2ARk

h

r:

Note that, k*(q) is decreasing in q. Hence, k*(q2i+1) P k*(q2i+2). On the other hand, for a given k, FCi(k,q) is convex in q and,momentarily assuming q as a continuous variable, FCi(k,q) is minimized at q*(k) where

q�ðkÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kðAi

D þ AR=kÞhðk� 1Þ þw

s:

Using these results, we propose the following procedure to solve the first subproblem:

Algorithm 2. Let k = bk*(q2i+2)c and k ¼ dk�ðq2iþ1Þe.

Step 0: Set FC�i ¼ 1 and k = k.Step 1: Find q*(k). If q*(k) R [q2i+1,q2i+2], set q*(k) to the closest boundary point.Step 2: Compute FCi(k,q*(k)). If FCiðk; q�ðkÞÞ < FC�i , then set FC�i ¼ FCiðk; q�ðkÞÞ.Step 2: Increase k by 1. If k 6 k, go to Step 1.

4.2.2. Analysis of LCi(k,q)The analysis is similar to that of FCi(k,q). For LCi(k,q),

k�ðqÞ ¼ 1q

ffiffiffiffiffiffiffiffiffiffiffi2ARk

h

r;

q�ðkÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kðAi

D þ AR=kÞhðk� 1Þ þw

s;

k ¼ bk�ðq2iþ1Þc; and�k ¼ dk�ðq2iÞ:e:

To solve the second subproblem, we simply implement Algorithm 2 with the updated values to minimize LCi(k,q).

4.2.3. SolutionAfter solving each subproblem, we simply pick the solution which yields the minimum expected cost as the optimal solu-

tion of C(k,q).

5. Concluding comments

Common carriage is widely used in logistics practice, especially in retail, grocery, and electronics industries. This type oftransportation is more desirable for the load sizes that are not as large to justify the use of full-truck-load (FTL) shipments.


However, common carrier charges reflect the scale economies, as well. Hence, given the option to manage the resupply deci-sions of the retailers’ orders, vendors may find it in their best interest to consolidate small orders into a large shipment loadwhen they use common carriage. Considering such an application arising in the context of a VMI arrangement, we developedmodels and solution procedures for finding the optimal inventory and outbound dispatch scheduling policy parameters forthe cases where the vendor implements time-based and quantity-based consolidation policies while using common carriagefor the outbound dispatches. Our results contribute to the literature by providing new analytical models for efficient VMIpractices under common carriage. From a methodological perspective, the resulting models lead to nonlinear optimizationproblems with non-differentiable cost functions. Our proposed algorithmic approaches offer methodological tools to obtainnumerical solutions for these problems.

The common carrier cost we used in our analysis includes a piecewise decreasing per unit rate. Our solution techniquescan be easily generalized to the case where this cost also includes a fixed component. A future area of research is to considerother transportation cost structures, e.g., incremental freight discounts, that demonstrate scale economies and investigatehow shipment consolidation policies are affected under various cost structures.

In logistics practice, many companies are using private carriage, contract carriage, and common carriage simultaneously.While larger loads and scheduled shipments are often made by private fleets, expedited orders and smaller loads are shippedvia contract or common carriers. An important research area is to investigate the combined use of various alternatives anddetermine the optimal policy parameters for how to consolidate shipments and how to select which alternative to use. Addi-tional potential areas of research include a careful analysis of shipment consolidation under other emerging practices such aschannel coordination (Chen and Chen, 2005) and cube-based pricing.

References

Abdelwahab, W.M., Sargious, M., 1990. Freight rate structure and optimal shipment size in freight transportation. Logistics and Transportation Review 6 (3),271–292.

Axsäter, S., 2001. A note on stock replenishment and shipment scheduling for vendor-managed-inventory. Management Science 47 (9), 1306–1310.Ballou, R.H., 2003. Business Logistics/Supply Chain Management. Prentice Hall.Burns, L.D., Hall, R.W., Blumenfeld, D.E., Daganzo, C.F., 1985. Distribution strategies that minimize transportation and inventory costs. Operations Research

33 (3), 469–490.Campbell, J.F., 1990. Designing logistics systems by analyzing transportation, inventory and terminal cost tradeoffs. Journal of Business Logistics 11 (1),

159–179.Çetinkaya, S., Bookbinder, J.H., 2003. Stochastic models for the dispatch of consolidated shipments. Transportation Research – Part B 37, 747–768.Çetinkaya, S., Lee, C.-Y., 2000. Stock replenishment and shipment scheduling for vendor managed inventory systems. Management Science 46 (2), 217–232.Çetinkaya, S., Mutlu, F., Lee, C.-Y., 2006. A comparison of outbound dispatch policies for vendor-managed inventory. European Journal of Operational

Research 171 (2), 1094–1112.Çetinkaya, S., Tekin, E., Lee, C.-Y., 2008. A stochastic model for integrated inventory replenishment and outbound shipment release decisions. IIE

Transactions 40 (8), 324–340.Chan, L.M.A., Muriel, A., Shen, Z.-J., Simchi-Levi, D., 2002. On the effectiveness of zero-inventory-ordering policies for the economic lot-sizing model with a

class of piecewise linear cost structures. Operations Research 50 (6), 1058–1067.Chen, T.H., Chen, J.M., 2005. Optimizing supply chain collaboration based on joint replenishment and channel coordination. Transportation Research – Part E

41, 261–285.Closs, D.J., Cook, R.L., 1987. Multi-stage transportation consolidation analysis using dynamic simulation. International Journal of Physical Distribution and

Materials Management 17 (3), 28–45.Cooper, M.C., 1984. Cost and delivery time implications of freight consolidation and warehouse strategies. International Journal of Physical Distribution and

Materials Management 14 (6), 47–67.Daganzo, C.F., 1988. Shipment composition enhancement at a consolidation center. Transportation Research – Part B 22 (2), 103–124.Dong, Y., Xu, K., Dresner, M., 2007. Environmental determinants of VMI adoption: an exploratory analysis. Transportation Research – Part E 43, 355–369.Gupta, Y.P., Bagchi, P.K., 1987. Inbound freight consolidation under just-in-time procurement: application of clearing models. Journal of Business Logistics 8

(2), 74–94.Hall, R.W., 1987. Consolidation strategy: inventory, vehicles, and terminals. Journal of Business Logistics 8 (2), 57–73.Higginson, J.K., 1993. Modeling shipper costs in physical distribution analysis. Transportation Research – Part A 27 (2), 113–124.Higginson, J.K., Bookbinder, J.H., 1994. Policy recommendations for a shipment consolidation program. Journal of Business Logistics 15 (1), 87–112.Higginson, J.K., Bookbinder, J.H., 1995. Markovian decision processes in shipment consolidation. Transportation Science 29 (3), 242–255.Jackson, G.C., 1981. Evaluating order consolidation strategies using simulation. Journal of Business Logistics 2 (2), 110–138.Masters, J.M., 1980. The effects of freight consolidation on customer service. Journal of Business Logistics 2 (1), 55–74.Mutlu, F., Çetinkaya, S., Bookbinder, J.H., 2010. An analytical nodel for computing the optimal time-and-quantity-based policy for consolidated shipments.

IIE Transactions 42 (5), 367–377.Pollock, T., 1978. A management guide to LTL consolidation. Traffic World 3, 29–35.Yao, Y., Dresner, M., 2008. The inventory value of information sharing, continuous replenishment, and vendor-managed inventory. Transportation Research

– Part E 44, 361–378.

Documents

An integrated model for stock replenishment and shipment scheduling under common carrier dispatch costs